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CAT 2021 Slot 2 — QA Question 15

IndicesEasy

For all possible integers n satisfying 2.25 ≤ 2 + 2n+2 ≤ 202, the number of integer values of 3 + 3n+1 is

Answer & solution

Answer: 7

Solution

Easy

First isolate 2 n+22^{\,n+2} from the given inequality to bound the integer nn. Then notice the question asks for the count of integer values of 3+3 n+13+3^{\,n+1}: this is an integer only when the exponent n+1≥0n+1\ge 0, so further restrict nn, and count.

1

Bound nn from the inequality. Subtract 22 throughout.

2.25≤2+2 n+2≤202⇒ 0.25≤2 n+2≤200\begin{aligned} &2.25\le 2+2^{\,n+2}\le 202\\ &\Rightarrow\ 0.25\le 2^{\,n+2}\le 200 \end{aligned}
2

Translate the powers of two into bounds on nn. Since 0.25=2−20.25=2^{-2} and 27=128≤200<256=282^{7}=128\le200<256=2^{8}.

2 n+2≥2−2 ⇒ n+2≥−2 ⇒ n≥−42 n+2≤200 ⇒ n+2≤7 ⇒ n≤5⇒ −4≤n≤5\begin{aligned} &2^{\,n+2}\ge 2^{-2}\ \Rightarrow\ n+2\ge -2\ \Rightarrow\ n\ge -4\\ &2^{\,n+2}\le 200\ \Rightarrow\ n+2\le 7\ \Rightarrow\ n\le 5\\ &\Rightarrow\ -4\le n\le 5 \end{aligned}
3

Require 3+3 n+13+3^{\,n+1} to be an integer. 3 n+13^{\,n+1} is an integer only when n+1≥0n+1\ge 0, i.e. n≥−1n\ge -1. Combining with step 2 gives −1≤n≤5-1\le n\le 5.

n∈{−1,0,1,2,3,4,5}⇒ number of values=7\begin{aligned} &n\in\{-1,0,1,2,3,4,5\}\\ &\Rightarrow\ \text{number of values}=7 \end{aligned}
77
CAT 2021 Slot 2 QA Q15: For all possible integers n satisfying 2.25 &le; 2 + 2 n+2 &le; 202, the number of integer values of 3 + 3 n+1 — Solution | TheCATExam